Negative Pell Equation Research Articles

The Integer Solution of the Negative Pell Equation Involving Palindrome Numbers

Abstract: The objective of the article is to find an integer solution that is non-zero for the negative Pell’s equation involving Palindrome number

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

An Integral Solution of Negative Pell Equation

Abstract: We look for non–trivial integer solution to the equation

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Работа посвящена изучению модели точечных вихрей, предложенной немецким ученым Германом Гельмгольцем. Найдены необходимые и достаточные условия существования бесконечного числа неэквивалентных статических конфигураций для системы, состоящей из двух точечных вихрей интенсивности $\Gamma_1$ и произвольного числа точечных вихрей интенсивности $\Gamma_2$. Проведена классификация таких конфигураций. Впервые обнаружена связь между отрицательным диофантовым уравнением Пелля и статическими конфигурациями точечных вихрей на плоскости. Библиография: 20 названий.

The 8-rank of the narrow class group and the negative Pell equation

Abstract Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$ . Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ is solvable with $x, y \in \mathbb {Z}$ is $58.1\%$ , to the nearest tenth of a percent. By studying the distribution of the $8$ -rank of narrow class groups $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$ , we prove that the infimum of this density is at least $53.8\%$ .

Redei reciprocity, governing fields and negative Pell

AbstractWe discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

Semicomplete Arithmetic Sequences, Division of Hypercubes, and the Pell Constant

In this paper we produce a few continuations of our previous work on partitions into fractions. Specifically, we study strictly increasing sequences of positive integers such that there are partitions for all natural numbers less than the floor of the sum of the first j terms divided by j, where j is greater than two. We also require that all summands be distinct terms drawn from this series of fractions. We call such sequences “semicomplete”. We find that there are only three semicomplete arithmetic sequences. We also study sequences that give the maximum number of pieces that an M dimensional hypercube can be cut into using N – 1 hyperplanes. We find that these are semicomplete in one, two, three, and four dimensions. As an aside, we use one of our generating functions to produce what appears to be a new identity for the Pell constant, a number which is closely connected to the density of solutions to the negative Pell equation.

On The Negative Pell Equation

On The Negative Pell Equation

Observation on the Negative Pell Equation

Observation on the Negative Pell Equation

On the double EPW sextic associated to a Gushel–Mukai fourfold

In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.

Spins of prime ideals and the negative Pell equation

Let$p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group$\text{Cl}(-4p)$of the imaginary quadratic number field$\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group$\text{Cl}(8p)$of the real quadratic field$\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation$x^{2}-2py^{2}=-1$over the integers; (iv) $2$-part of the Tate–Šafarevič group$\unicode[STIX]{x0428}(E_{p})$of the congruent number elliptic curve$E_{p}:y^{2}=x^{3}-p^{2}x$. Our results are conditional on a standard conjecture about short character sums.

ON THE NEGATIVE PELL EQUATION y^2=90*x^2-9

The negative pell equation represented by the binary quadratic equation is analysed for its non-zero distinct integer solutions. A few interesting relations among the solutions are presented. Employing the solutions of the equation under consideration, the integer solutions for a few choices of hyperbola and parabola are obtained.

NOTES ON QUADRATIC INTEGERS AND REAL QUADRATIC NUMBER FIELDS

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_{d}$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_{d} \gg (\log d)^{2}$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^{2} - d Y^{2} = -1$ (or more generally $X^{2} - D Y^{2} = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $\mu$ are principal.

English

English

On the negative Pell equation

Let e be the fundamental unit of a field $$ Q\left( {\sqrt {d} } \right) $$ . In the paper, it is proved that e > d 3/2/ log2 d for almost all d such that N(e) = −1. Bibliography: 6 titles.

On the negative Pell equation

We give asymptotic upper and lower bounds for the number of squarefree d (0 < d ≤ X) such that the equation x 2 - dy 2 = -1 is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm -1 and give strong evidence in the direction of a conjecture due to P. Stevenhagen.

On the matrix negative Pell equation

Let N be a set of natural numbers and Z be a set of integers. Let M2(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X − dY 2 = −I with d ∈ N for nonsingular X, Y belonging to M2(Z) and his generalization (Pn) n ∑

Power Residue Criteria for Quadratic Units and the Negative Pell Equation

AbstractLet d &gt; 1 be a square-free integer. Power residue criteria for the fundamental unit εd of the real quadratic fields modulo a prime p (for certain d and p) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation x2 − dp2y2 = −1. The most important solvability criterion deals with all d for which has an elementary abelian 2-class group and p ≡ 5 (mod 8) or p ≡ 9 (mod 16).

The negative Pell equation and Pythagorean triples

We give a simple criterion for the solvability in integers of the negative Pell equation $x^2 - dy^2 = -1$ by means of primitive Pythagorean triples. The method of proof allows us to also solve the problem of determinig the fundamental units of negative norm in real quadratic fields.

Density computations for real quadratic units

In order to study the density of the set of positive integers d d for which the negative Pell equation x 2 − d y 2 = − 1 x^{2}-dy^{2}=-1 is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in log ⁡ d \log d as the factorization of d d is a natural part of the input for the values of d d we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion P = 1 − ∏ j ≥ 1 o d d ( 1 − 2 − j ) ≈ .58 P = 1 - \prod _{j\ge 1\; \mathrm {odd}} (1-2^{-j}) \approx .58 of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm − 1 -1 .